1.0 Basic Interferometry
Interferometry involves the analysis of interfering waves in order to measure a distance. Interferometers, which are measurement tools that perform interferometry, typically reflect a first series of optical waves from a first reflecting surface; and, reflect a second series of optical waves from a second reflecting surface. The first and second series of waves are subsequently combined to form a combined waveform. A signal produced through the detection of the combined waveform is then processed to understand the relative positioning of the reflective surfaces. FIG. 1 shows an embodiment of a type of interferometer that is often referred to as a Michelson interferometer.
Referring to FIG. 1, a light source 101 and splitter 102 are used to form a first group of light waves that are directed to a reference mirror 104; and, a second group of light waves that are directed to a plane mirror 103. The splitter 102 effectively divides the light 106 from the light source 101 in order to form these groups of light waves. Typically, the splitter 102 is designed to split the light 106 from the light source 101 evenly so that 50% of the optical intensity from the light source 101 is directed to the reference mirror 104 and 50% of the optical intensity from the light source 101 is directed to the plane mirror 103.
At least a portion of the light that is directed to the plane mirror 103 reflects back to the splitter 102 (by traveling in the +z direction after reflection); and, at least a portion of the light that is directed to the reference mirror 104 reflects back to the splitter 102 (by traveling in the −y direction after reflection). The reflected light from the reference mirror 104 and plane mirror 103 are effectively combined by the splitter 102 to form a third group of light waves that propagate in the −y direction and impinge upon a detector 105. The optical intensity pattern(s) observed by the detector 105 are then analyzed in order to measure the difference between the distances d1, d2 that exist between the plane mirror 103 and the reference mirror 104, respectively.
That is, for planar wavefronts, if distance d2 is known, distance d1 can be measured by measuring the intensity of the light received at the detector 105. Here, according to wave interference principles, if distance d1 is equal to distance d2; then, the reflected waveforms will constructively interfere with one another when combined by the splitter 102 (so that their amplitudes are added together). Likewise, if the difference between distance d1 and distance d2 is one half the wavelength of the light emitted by light source 101; then, the reflected waveforms will destructively interfere with one another when combined by the splitter 102 (so that their amplitudes are subtracted from one another).
The former situation (constructive interference) produces a relative maximum optical intensity (i.e., a relative “brightest” light) at the detector 105; and, the later situation (destructive interference) produces a relative minimum optical intensity (i.e., a relative “darkest” light). When the difference between distance d1 and distance d2 is somewhere between zero and one half the wavelength of the light emitted by the light source 101, the intensity of the light that is observed by the detector 105 is less than the relative brightest light from constructive interference but greater than the relative darkest light from destructive interference (e.g., a shade of “gray” between the relative “brightest” and “darkest” light intensities). The precise “shade of gray” observed by the detector 105 is a function of the difference between distance d1 and distance d2.
In particular, the light observed by the detector 105 becomes darker as the difference between distance d1 and distance d2 depart from zero and approach one half the wavelength of the light emitted by the light source 101. Thus, the difference between d1 and d2 can be accurately measured by analyzing the optical intensity observed by the detector 105. For planar optical wavefronts, the optical intensity should be “constant” over the surface of the detector 105 because (according to a simplistic perspective) whatever the difference between distance d1 and d2 (even if zero), an identical “effect” will apply to each optical path length experienced by any pair of reflected rays that are combined by the splitter 102 to form an optical ray that is directed to the detector 105.
Here, note that the 45° orientation of the splitter 102 causes the reference mirror directed and plane mirror directed portions of light to travel equal distances within the splitter 102. For example, analysis of FIG. 1 will reveal that the reference mirror and plane mirror directed portions of ray 107 travel equal distances within splitter 102; and, that the reference mirror and plane mirror directed portions of ray 108 travel equal distances within splitter 102. As all light rays traveling to detector 105 from splitter 102 must travel the same distance d3, it is clear then that the only difference in optical path length as between the plane mirror and reference mirror directed portions of light (that are combined to form a common ray that impinges upon the detector 105), must arise from a difference between d1 and d2; and; likewise, for planar wavefronts, a difference between d1 and d2 should affect all light rays impingent upon detector 105 equally. As such, ideally, the same “shade of gray” should be observed across the entirety of the detector; and, the particular “shade of gray” can be used to determine the difference between distance d1 and d2 from wave interference principles.
2.0 Interferometer Having a “Tilted” Reference Mirror
Referring to FIG. 2, when the reference mirror 204 is tilted (e.g., such that θ is greater than 0° as observed in FIG. 2), the optical intensity observed at the detector 205 departs from being uniform across the surface of the detector 205 because the differences in optical path length as between plane mirror 203 and reference mirror 204 directed portions of light are no longer uniform. Better said, the “tilt” in the reference mirror 204 causes variation in optical path length amongst the light waves that are directed to the reference mirror 204; which, in turn, causes variation in the optical intensity observed at the detector 205.
Here, as wave interference principles will still apply at the detector 205, the variation in optical path length that is introduced by the tilted reference mirror 204 can be viewed as causing optical path length differences experienced by light that impinges upon the detector 202 to effectively progress through distances of λ/2, λ, 3λ/2, 2λ, 5λ/2, 3λ, etc. (where A is the wavelength of the light source). This, in turn, corresponds to continuous back and forth transitioning between constructive interference and destructive interference along the z axis of the detector 205. FIG. 3a shows an example of the optical intensity pattern 350 observed at the detector 305 when the reference mirror of an interferometer is tilted (as observed in FIG. 2).
Here, notice that the optical intensity pattern 350 includes relative minima 352a, 352b, and 352c; and, relative maxima 351a, 351b, 351c, and 351d. The relative minima 352a, 352b, and 352c, which should appear as a “darkest” hue within their region of the detector 305, are referred to as “fringe lines”. FIG. 3b shows a depiction of the fringe lines that appear on a detector when the reference mirror of an interferometer is tilted. Here, ideally, fringe lines that run along the x axis will repeatedly appear as one moves across the z axis of the detector. The separation of the fringe lines is a function of both the wavelength of the light source and the angle at which the reference mirror is tilted. More specifically, the separation of the fringe lines is proportional to the wavelength of the light source and inversely proportional to the angle of the tilt. Hence, fringe line separation may be expressed as ˜λ/θ.